[Q] Is a multinomial model appropriate for these data?

I’m trying to understand if multinomial models are appropriate for my analysis. To give a simplified example, let’s say we’re interested in the choice of someone randomly picking coloured balls out of a hat. If there are 10 green, 10 red and 10 orange balls, then the baseline expected probability of picking any colour is equal.

But if there are unequal number of coloured balls, e.g. 25 green, 3 red and 2 orange, this is going to affect the probability of picking a ball. You’re more likely to pick a green, simply because there’s far more of them.

To add complexity, we repeat this choice exercise for multiple different bags, all with varying numbers of coloured balls . So there would presumably need to be some sort of random term .

Can multinomial models account for

* unequal numbers of coloured balls in a bag
* multiple bags, all with different number of coloured balls in them

I’ve been looking to analyse my data in R, with the nnet and mlogit packages but I’m not sure if multinomial models are actually appropriate or not.

Thanks for the help!